Taylor-type 1-step-ahead numerical differentiation rule for first-order derivative approximation and ZNN discretization

Abstract

h i g h l i g h t s• A formula is proposed to approximate the first-order derivative.• An optimal step length rule for the proposed formula is investigated. • A Taylor-type ZNN model is derived for time-varying matrix inversion. a b s t r a c tIn order to achieve higher computational precision in approximating the first-order derivative and discretize more effectively the continuous-time Zhang neural network (ZNN), a Taylor-type numerical differentiation rule is proposed and investigated in this paper. This rule not only greatly remedies some intrinsic weaknesses of the backward and central numerical differentiation rules, but also overcomes the limitation of the Lagrange-type numerical differentiation rules in ZNN discretization. In addition, a formula is proposed to obtain the optimal step-length of the Taylor-type numerical differentiation rule. Moreover, based on the proposed numerical differentiation rule, the stability, convergence and residual error of the Taylor-type discrete-time ZNN (DTZNN) are analyzed. Numerical experimental results further substantiate the efficacy and advantages of the proposed Taylor-type numerical differentiation rule for first-order derivative approximation and ZNN discretization.

New formula of 4-instant g-square finite difference (4IgSFD) applied to time-variant matrix inversion

“…For the proposed 4IgSFD-type DTZD model (3), the following definitions (i.e., Definitions 1 through 3) [12] and theorems (i.e., Theorems 1 and 2) are presented to guarantee its stability and convergence.…”

“…However, as known, the backward difference may not adapt to the fast variational rate of the first-order derivative of the target point, and the numerical difference is urgently needed to be inversely applied to the discretization of continuous-time ZD model. Moreover, a finite difference formula does not necessarily generate a stable and convergent discrete-time ZD (DTZD) model [12]. Considering the aforementioned situations, in this paper, we propose a new formula of 4IgSFD to obtain the first-order derivative approximation and discretize the continuous-time ZD model for the time-variant matrix inversion.…”

“…Note that, because the continuous-time ZD model may be difficult for digital computer to realize directly, many finite difference methods have been considered to obtain the first-order derivative approximation and then discretize the continuous-time ZD model for time-variant matrix inversion, such as the polynomial interpolation method [11] This work is supported by the 2012 Scholarship Award for Excellent Doctoral Student Granted by Ministry of Education of China (with number 3191004), and by the Foundation of Key Laboratory of Autonomous Systems and Networked Control of Ministry of Education of China (with number 2013A07). as well as the method of Lagrange-type 1-step-ahead numerical differentiation (including Euler forward-difference rule) [12]. However, as known, the backward difference may not adapt to the fast variational rate of the first-order derivative of the target point, and the numerical difference is urgently needed to be inversely applied to the discretization of continuous-time ZD model.…”

General Six-Step Discrete-Time Zhang Neural Network for Time-Varying Tensor Absolute Value Equations

“…Notably, the di erence between the stability criterion proposed by Routh and Hurwitz and those in [16,17] mainly lie in: (1) they are proposed for di erent systems; (2) the former is a su cient and necessary condition, while the latter are only su cient conditions. e following is the de nition of the Taylor-type 1-stepahead numerical di erentiation rule for the rst-order derivative approximation, which is o en termed as Zhang et al discretization (ZeaD) formula [18].…”

The Eight Epochs of Math as Regards Past and Future Matrix Computations

“…Consider the discrete time-varying generalized matrix inverse problemB k+1 −Y + k+1 = 0, with B k = sin(0.5t k ) cos(0.1t k ) − sin(0.1t k ) − cos(0.1t k ) sin(0.1t k ) cos(0.1t k ). Typical residual errors generated by the 5 instants , the 4 instants and the Euler formulas with different sampling gaps τ when solving the discrete time-varying generalized-matrix-inverse problem(29) for t end = 30 s, and h = 0.1. Here we consider the discrete time-varying matrix inversion problemA k+1 X k+1 = I with A k = sin(0.5t k ) + 2 cos(0.5t k ) cos(0.5t k ) sin(0.5t k ) + 2 .Profiles of the (1,1) entry X 1,1 Profiles of the (1,2) entry X 1,2 Profiles of the (2,2) entry X 2,Profiles of the four entries of the solution X when solving the discrete time-varying matrix-inverse problem (30) with τ = 0.1 s. Here the solid curves show the solution entries generated by the 5 instants discrete model obtained from random starting values and the dash-dotted curves depict the theoretical solutions.…”